Chapter 2 Discrete Fourier Transform DFT Topics Discrete

Chapter 2 Discrete Fourier Transform (DFT) Topics: Ø Discrete Fourier Transform. • • • Using the DFT to Compute the Continuous Fourier Transform. Comparing DFT and CFT Using the DFT to Compute the Fourier Series Huseyin Bilgekul Eeng 360 Communication Systems I Department of Electrical and Electronic Engineering Eastern Mediterranean University 1

Discrete Fourier Transform (DFT) ØDefinition: The Discrete Fourier Transform (DFT) is defined by: Where n = 0, 1, 2, …. , N-1 The Inverse Discrete Fourier Transform (IDFT) is defined by: where k = 0, 1, 2, …. , N-1. The Fast Fourier Transform (FFT) is a fast algorithm for evaluating the DFT. 2

Using the DFT to Compute the Continuous Fourier Transform Ø Suppose the CFT of a waveform w(t) is to be evaluated using DFT. 1. The time waveform is first windowed (truncated) over the interval (0, T) so that only a finite number of samples, N, are needed. The windowed waveform ww(t) is 2. The Fourier transform of the windowed waveform is 3. Now we approximate the CFT by using a finite series to represent the integral, 4. t = k∆t, f = n/T, dt = ∆t, and ∆t = T/N 3

Computing CFT Using DFT • We obtain the relation between the CFT and DFT; that is, f = n/T and ∆t = T/N • The sample values used in the DFT computation are x(k) = w(k∆t), • If the spectrum is desired for negative frequencies – the computer returns X(n) for the positive n values of 0, 1, …, N-1 – It must be modified to give spectral values over the entire fundamental range of -fs/2 < f <fs/2. For positive frequencies we use For Negative Frequencies 4

Comparison of DFT and the Continuous Fourier Transform (CFT) Relationship between the DFT and the CFT involves three concepts: • Windowing, • Sampling, • Periodic sample generation 5

Comparison of DFT and the Continuous Fourier Transform (CFT) Relationship between the DFT and the CFT involves three concepts: • Windowing, • Sampling, • Periodic sample generation 6

Fast Fourier Transform Ø The Fast Fourier Transform (FFT) is a fast algorithm for evaluating DFT. Block diagrams depicting the decomposition of an inverse DTFS as a combination of lower order inverse DTFS’s. (a) Eight-point inverse DTFS represented in terms of two fourpoint inverse DTFS’s. (b) four-point inverse DTFS represented in terms of two-point inverse DTFS’s. (c) Two-point inverse DTFS. 7

Using the DFT to Compute the Fourier Series Ø The Discrete Fourier Transform (DFT) may also be used to compute the complex Fourier series. Ø Fourier series coefficients are related to DFT by, Ø Block diagram depicting the sequence of operations involved in approximating the FT with the DTFS. 8

Ex. 2. 17 Use DFT to compute the spectrum of a Sinusoid 9

Ex. 2. 17 Use DFT to compute the spectrum of a Sinusoid Spectrum of a sinusoid obtained by using the MATLAB DFT. 10

Using the DFT to Compute the Fourier Series The DTFT and length-N DTFS of a 32 -point cosine. The dashed line denotes the CFT. While the stems represent N|X[k]|. (a) N = 32 (b) N = 60 (c) N = 120. 11

Using the DFT to Compute the Fourier Series The DTFS approximation to the FT of x(t) = cos(2 (0. 4)t) + cos(2 (0. 45)t). The stems denote |Y[k]|, while the solid lines denote CFT. (a) M = 40. (b) M = 2000. (c) Behavior in the vicinity of the sinusoidal frequencies for M = 2000. (d) Behavior in the vicinity of the sinusoidal frequencies for M = 2010 12